Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation

We consider a Cauchy problem for a semilinear heat equation ut = u +up in RN ×(0,T ), u(x, 0) = u0(x) 0 inRN with p >pS where pS is the Sobolev exponent. If u(x, t) = (T −t)−1/(p−1)ϕ((T −t)−1/2x) for x ∈ RN and t ∈ [0,T ), where ϕ is a regular positive solution of ϕ − y 2∇ϕ − 1 p −1 ϕ + ϕp =0 inRN, (P) then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ ≡ (p −1)−1/(p−1) for all p >1. Let pL be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for pS < p < pL. We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p >pL. © 2009 Elsevier Inc. All rights reserved